Question: Here's a partially-filled Hessian matrix. $\begin{bmatrix} ??? & -xy\sin(xy) \\ \\ -xy\sin(xy) & -x^2\sin(xy) \end{bmatrix}$ What is the missing entry? Choose 1 answer: Choose 1 answer: (Choice A) A $-y^2\sin(xy)$ (Choice B) B $y^2\sin(xy)$ (Choice C) C $2 - x^2\sin(xy)$ (Choice D) D There's not enough information.
Answer: The Hessian of a scalar field $f$ is the matrix that contains all its second-order partial derivative information. $\bold{H}(f) = \begin{bmatrix} f_{xx} & f_{xy} \\ \\ f_{yx} & f_{yy} \end{bmatrix}$ Because the order of mixed partial derivatives often doesn't matter, the Hessian matrix is usually symmetric. The symmetry of the Hessian means that many elements are repeated, but all the diagonal entries are only given once. Therefore, we don't have enough information to determine $f_{xx}$. [But don't we know the mixed partial derivatives? Isn't that enough information?]